Linear Least Squares Fitting Technique  Practical Electron Microscopy and Database   An Online Book  

Microanalysis  EM Book http://www.globalsino.com/EM/  


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Linear least squares fitting (LLSF) technique is the simplest and most commonly used form of linear regression and provides a solution to a problem of finding the best fitting straight line through a set of points. There are many applications with this technique, for instance, in the quantifications in electron and photoninduced Xrays, linear leastsquare fitting can be
used to deconvolute the overlapped peaks once the continuum background is eliminated by a proper technique. In the LLSF method, it is assumed that each channel of the measured spectrum is equal to the sum of a set of the reference spectra in the
same channel, as given below, The reference spectra may be any set of curves that can be extracted from a proper model, including the information of the shape and amplitude of the particular Xray lines of interest. In general, such references are a set of measured spectra from the pure elements 1, 2, ... n. According to the leastsquares method, the best set of the A_{i} coefficients are obtained by minimizing the value of x^{2}, which is given by, In ideal leastsquares fitting, if the x^{2} is normalized (dividing by the number of fitted points minus degrees of freedom), it should be approximately one in average. Values of x^{2} that are much greater than one indicates some systematic error in the fit. Different from the ideal case described in Equation 1760b, the practical leastsquares fitting must incorporate a treatment of the continuum differences. [1] The procedure of fitting the spectrum from a sample using LLSF technique is:
[1] McCarthy, J.J., and F.H. Schamber, 1981, LeastSquares Fit with Digital Filter: A Status Report. In Energy Dispersive Xray Spectrometry, edited by K.F.J. Heinrich, D.E. Newbury, R.L. Myklebust, and C.E. Fiori, pp. 273296. National Bureau of Standards Special Publication 604, Washington, D.C.


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